Difference between revisions of "Floating Elastic Plate"

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= Introduction =
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This page has moved to [[:Category:Floating Elastic Plate|Floating Elastic Plate]].
  
The floating elastic plate is one of the best studied problems in hydroelasticity. It can be used to model a range of
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Please change the link to the new page
physical structures such as a floating break water, an ice floe or a [[VLFS]]). The equations of motion were formulated
 
more than 100 years ago and a discussion of the problem appears in [[Stoker_1957a|Stoker 1957]]. The problem can
 
be divided into the two and three dimensional formulations which are closely related. The plate is assumed to be isotropic while the water motion is irrotational and inviscid.
 
 
 
= Two Dimensional Problem =
 
 
 
== Equations of Motion ==
 
 
 
When considering a two dimensional problem, the <math>y</math> variable is dropped and the plate is regarded as a beam. There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the [[Bernoulli-Euler Beam]] which is commonly used in the two dimensional hydroelastic analysis. Other beam theories include the [[Timoshenko Beam]] theory and [[Reddy-Bickford Beam]] theory where shear deformation of higher order is considered.
 
 
 
 
 
For a Bernoulli-Euler beam on the surface of the water, the equation of motion is given
 
by the following
 
 
 
<math>D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = p</math>
 
 
 
where <math>D</math> is the flexural rigidity, <math>\rho_i</math> is the density of the beam,
 
<math>h</math> is the thickness of the beam (assumed constant), <math> p</math> is the pressure
 
and <math>\eta</math> is the beam vertical displacement.
 
 
 
The edges of the plate satisfy the natural boundary condition (i.e. free-edge boundary conditions).
 
 
 
<math>\frac{\partial^2 \eta}{\partial x^2} = 0, \,\,\frac{\partial^3 \eta}{\partial x^3} = 0</math>
 
 
 
at the edges of the plate.
 
 
 
The pressure is given by the linearised Bernoulli equation at the wetted surface (assuming zero
 
pressure at the surface), i.e.
 
 
 
<math>p = \rho g \frac{\partial \phi}{\partial z} + \rho \frac{\partial \phi}{\partial t}</math>
 
 
 
where <math>\rho</math> is the water density and <math>g</math> is gravity, and <math>\phi</math>
 
is the velocity potential. The velocity potential is governed by Laplace's equation through out
 
the fluid domain subject to the free surface condition and the condition of no flow through the
 
bottom surface. If we denote the beam-covered (or possible multiple beams covered) region of the fluid by <math>P</math> and the free surface by <math>F</math> the equations of motion for the
 
[[Frequency Domain Problem]] with frequency <math>\omega</math> for water of
 
[[Finite Depth]] are the following. At the surface
 
we have the dynamic condition
 
 
 
<math>D\frac{\partial^4 \eta}{\partial x^4} +\left(\rho g- \omega^2 \rho_i h \right)\eta =
 
i\omega \rho \phi, \, z=0, \, x\in P</math>
 
 
 
<math>0=
 
\rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, x\in F</math>
 
 
 
and the kinematic condition
 
 
 
<math>\frac{\partial\phi}{\partial z} = i\omega\eta</math>
 
 
 
 
 
 
 
The equation within the fluid is governed by [[Laplace's Equation]]
 
 
 
<math>\nabla^2\phi =0 </math>
 
 
 
and we have the no-flow condition through the bottom boundary
 
 
 
<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>
 
 
 
(so we have a fluid of constant depth with the bottom surface at <math>z=-h</math> and the
 
free surface or plate covered surface are at <math>z=0</math>).
 
<math> g </math> is the acceleration due to gravity,  <math> \rho_i </math> and <math> \rho </math>
 
are the densities of the plate and the water respectively, <math> h </math> and <math> D </math>
 
the thickness and flexural rigidity of the plate.
 
 
 
== Solution Methods ==
 
 
 
There are many different methods to solve the corresponding equations ranging from highly analytic such
 
as the [[Wiener-Hopf]] to very numerical based on [[Eigenfunction Matching Method]] which are
 
applicable and have advantages in different situations. We describe here some of the solutions
 
which have been developed grouped by problem
 
 
 
=== Single Crack ===
 
 
 
The simplest problem to consider is one where there are only two semi-infinite plates of identical properties separated by a crack. A related problem in acoustics was considered by [[Kouzov_1963a|Kouzov 1963]] who used an integral representation of the problem to solve it explicitly using the Riemann-Hilbert technique. Recently the crack problem has been considered by [[Squire_Dixon_2000A|Squire and Dixon 2000]] and [[Williams_Squire_2002A|Williams and Squire 2002]] using a Green function method applicable to infinitely deep water and they obtained simple expressions for the reflection and transmission coefficients. [[Squire_Dixon_2001a|Squire and Dixon 2001]] extended the single crack problem to a multiple crack problem in which the semi-infinite regions are separated by a region consisting of a finite number of plates of finite size with all plates having identical properties. [[Evans_Porter_2005a|Evans and Porter 2005]] further considered the multiple crack problem for finitely deep water and provided an explicit solution.
 
 
 
The solution of [[Evans_Porter_2005a|Evans and Porter 2005]] expresses the potential
 
<math>\phi</math> in terms of a linear combination of the incident wave and certain source functions located at each of the cracks.
 
Along with satisfying the field and boundary conditions, these source functions satisfy the jump conditions in the displacements and gradients across each crack.
 
We will briefly present the solution of [[Evans_Porter_2005a|Evans and Porter 2005]]. They first define
 
<math>\chi(x,z)</math> to be the Two-Dimensional solution to the [[Free-Surface Green Function for a Floating Elastic Plate]]
 
given by
 
 
 
<math>
 
\chi(x,z) = -i\sum_{n=-2}^\infty\frac{\sin{(k(n)h)}\cos{(k(n)(z-h))}}{2\alpha C_n}e^{-\kappa(n)|x|},\,\,\,(1){eq:chi4}
 
</math>
 
 
 
where
 
 
 
<math>
 
C_n=\frac{1}{2}\left(h + \frac{(5\beta k(n)^4 + 1 - \alpha\gamma)\sin^2{(k(n)h)}}{\alpha}\right),
 
</math>
 
 
 
and <math>k(n)</math> are the solutions of the [[Dispersion Relation for a Floating Elastic Plate]]. 
 
 
 
Consequently, the source functions for a single crack at <math>x=0</math> can be defined as
 
 
 
<math>
 
\psi_s(x,z)= \beta(\chi_{xx}(x,z) - \nu k_y^2\chi(x,z)),\,\,\,
 
\psi_a(x,z)= \beta(\chi_{xxx}(x,z) - \nu_1 k_y^2\chi_x(x,z)),\,\,\,(2)
 
</math>
 
 
 
where <math>\nu_1 = 2-\nu</math>.
 
 
 
It can easily be shown that <math>\psi_s</math> is symmetric about <math>x = 0</math> and
 
<math>\psi_a</math> is antisymmetric about <math>x = 0</math>.
 
 
 
Substituting (1){eq:chi4} into (2){eq:psi1} gives
 
 
 
<math>
 
\psi_s(x,z)=
 
{
 
-\frac{\beta}{\alpha}
 
\sum_{n=-2}^\infty
 
\frac{g_n\cos{(k(n)(z+h))}}{2k_{xn}C_n}e^{\kappa_{n}|x|} },
 
\psi_a(x,z)=
 
{
 
{\rm sgn}(x) i\frac{\beta}{\alpha}\sum_{n=-2}^\infty
 
\frac{g_n'\cos{(k(n)(z+h))}}{2k_{xn}C_n}e^{\kappa_{n}|x|}},
 
</math>
 
 
 
where
 
 
 
<math>
 
g_n = -i\kappa(n)(-\kappa(n)^2 + \nu k_y^2)(\sin{(k(n)h)},
 
g'_n= \kappa(n)^2(-\kappa(n)^2 + \nu k_y^2)(\sin{(k(n)h)}.
 
</math>
 
 
 
We then express the solution to the problem as a linear combination of the
 
incident wave and pairs of source functions at each crack,
 
 
 
<math>
 
\phi(x,z) =
 
{ Ie^{-\kappa_{1}(0)(x-r_1)}\frac{\cos(k_1(0)(z+h))}{\cos(k_1(0)h)} }
 
+ \sum_{n=1}^{\Lambda-1}(P_n\psi_s(x-r_n,z) + Q_n\psi_a(x-r_n,z))\,\,\,(3)
 
</math>
 
 
 
where <math>P_n</math> and <math>Q_n</math> are coefficients to be solved which represent the jump in the gradient
 
and elevation respectively of the plates across the crack <math>x = a_j</math>.
 
The coefficients <math>P_n</math> and <math>Q_n</math> are found by applying the edge conditions  and to
 
the <math>z</math> derivative of <math>\phi</math> at <math>z=0</math>.
 
 
 
The reflection and transmission coefficients, <math>R_1(0)</math> and <math>T_\Lambda(0)</math> can be found from (3)
 
by taking the limits as <math>x\rightarrow\pm\infty</math> to obtain
 
 
 
<math>
 
R_1(0) e^{-\kappa_(0)r_1}= {- \frac{\beta}{\alpha}\sum_{n=1}^{\Lambda-1}
 
\frac{e^{\kappa(0)r_n}}{2k_0C_0}(g'_0Q_n + ig_0P_j)}
 
T_\Lambda(0) e^{\kappa(0)l_\Lambda}= 1 + {\frac{\beta}{\alpha}\sum_{n=1}^{\Lambda -1}\frac{e^{-\kappa(0)r_n}}{2k_0C_0}(g'_0Q_n - ig_0P_j)}
 
</math>
 
 
 
=== Two Semi-Infinite Plates of Different Properties ===
 
 
 
The next most simple problem is two semi-infinite plates of different properties. Often one of
 
the plates is taken to be open water which makes the problem simpler. In general, the solution method
 
developed for open water can be extended to two plates of different properties, the exception to
 
this is the [[Residue Calculus]] solution which applies only when one of the semi-infinite regions
 
is water.
 
 
 
====[[Wiener-Hopf]]====
 
 
 
The solution to the problem of two semi-infinite plates with different properties can be
 
solved by the Wiener-Hopf method. The first work on this problem was by [[Evans_Davies_1968a|Evans and Davies 1968]]
 
but they did not actually develop the method sufficiently to be able to calculate the solution.
 
The explicit solution was not found until the work of ...
 
 
 
====[[Eigenfunction Matching Method]]====
 
 
 
====[[Residue Calculus]]====
 
 
 
= Three Dimensional Problem =
 
 
 
== Equations of Motion ==
 
 
 
For a classical thin plate, the equation of motion is given by
 
 
 
<math>D\nabla ^4 w + \rho _i h w = p</math>
 

Latest revision as of 23:56, 15 June 2006

This page has moved to Floating Elastic Plate.

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